This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: r) and Y(s) in Figure 3.55 Block diagram for
Problem 3.22 Figure 3.56
Circuit for Problem 3.23 Figure 3.57 Unity feedback system
for Problem 3.24 Problems 159 Problems for Section 3.3: Effect of Pole Locations 3.23 For the electric circuit shown in Fig. 3.56, ﬁnd the following:
(a) The time—domain equation relating [(r) and 121(1);
(b) The limedomain equation relating i(t) and v20);
Vol (c) Assuming all initial conditions are zero, the transfer function 1;) and the damping
ratio 4 and undamped natural frequency w" of the system; ' (d) The values of R that will result in v20) having an overshoot of no more than 25%,
assuming v10) is a unit step, L = 10 mH. and C = 4 #F. L R mm It!) C up(l) _ i L 3.24 For the unity feedback system shown in Fig. 3.57. specify the gain K of the proportional
controller so that the output y(t) has an overshoot of no more than 10% in response to a unit step. 3.25 For the unity feedback system shown in Fig. 3.58, specify the gain and pole location
of the compensator so that the overall closedloop response to a unitstep input has an
overshoot of no more than 25%. and a 1% settling time of no more than 0.1 sec. Verify
your design using MATLAB. 160 Chapter3 Dynamic Response Figure 3.58 Unity feedback system
for Problem 3.25 Figure 3.59 Unity feedback system
for Problem 3.28 Figure 3.60 Desired closedloop
pole locations for
Problem 3.28 Problemsfor Section 3.4: TimeDomain Speciﬁcation 3.26 3.27 3.28 Suppose you desire the peak time of a given secondorder system to be less than 1”,
Draw the region in the splane that corresponds to values of the poles that meet the speciﬁcation (p < 4,. A certain servomechanism system has dynamics dominated by a pair of complex poles
and no ﬁnite zeros. The timedomain speciﬁcations on the rise time (t,), percent
overshoot (Mp), and settling time ([5) are given by t, 5 0.6 sec ,
Mp 5 l7%,
Is 5 9.2 sec. (a) Sketch the region in the splane where the poles could be placed so that the system
will meet all three speciﬁcations. (b) Indicate on your sketch the speciﬁc locations (denoted by x) that will have the
smallest risetime and also meet the settling time speciﬁcation exactly. Suppose you are to design a unity feedback controller for a ﬁrstorder plant depicted in Fig. 3.59. (As you will learn in Chapter 4, the conﬁguration shown is referred to as a proportionalintegral controller.) You are to design the controller so that the closedloop poles lie within the shaded regions shown in Fig. 3.60. (a) What values of w" and 1; correspond to the shaded regions in Fig. 3.59? (A simple
estimate from the ﬁgure is sufﬁcient.) (b) Let Ka‘ = a = 2. Find values for K and K, so that the poles of the closedloop
system lie within the shaded regions Problems 161 (c) Pr0ve that no matter what the values of K0 and or are, the controller provides enough
ﬂexibility to place the poles anywhere in the complex (left—halt) plane. 3.29 The openloop transfer function of a unity feedback system is K
s(s+2)' 0(3) : The desired system response to a step input is speciﬁed as peak time t,, = 1 sec and
overshoot M,, = 5%. (:1) Determine whether both speciﬁcations can be met simultaneously by selecting the
right value of K. (b) Sketch the associated region in the splane where both Specifications are met, and
indicate what root locations are possible for some likely values of K. (c) Relax the speciﬁcations in part (a) by the same factor and pick a suitable value for
K . and use MATLAB to verify that the new speciﬁcations are satisﬁed. 3.30 The equations of motion for the DC motor shown in Fig. 2.32 were given in Eqs. (2.52— 2.53) as
.. K K . K
JQO ‘i' (b + 0m : [T1471 ll (1 Assume that J,,, = 0.01 kgmz. b = 0.001 Nmscc,
Kc = 0.02 Vsec,
K, = 0.02 Nm/A, a = 10 S2. (3) Find the transfer function between the applied voltage Va and the motor speed 9,". (b) What is the steady—state speed of the. motor after a voltage V(( = 10 V has been
applied? . (c) Find the transfer function between the applied voltage v“ and the shaft angle 6,". (d) Suppose feedback is added to the system in part (c) so that it. becomes a position
servo device such that the applied voltage is given by "a = Kigr _ am) Where K is the feedback gain. Find the transfer function between 6,. and 9m. (9) What is the maximum value of K that can be used if an overshoot Mp < 20% is
desired? (f) What values of K will provide a rise time of less than 4 sec? (Ignore the Mp
constraint.) (g) Use MATLAB to plot the step response of the position sen/o system for values of
the gain K = 0.5, 1, and 2. Find the overshoot and rise time for each of the three step
responses by examining your plots. Are the plots consistent with your calculations
in parts (e) and (f)? 3.31 You wish to control the elevation of the satellitetracking antenna shown in Figs. 3.61
and 3.62. The antenna and drive parts have a moment of inertia J and a damping B; 162 Chapter 3 Dynamic Response Figure 3.61 Satellite—tracking
antenna Source: Courtesy Space
Systems/Lora! Figure 3.62 Schematic ofantenna
for Problem 3.31 these arise to some extent from bearing and aerodynamic friction, but mostly from the
back emf of the DC drive motor. The equations of motion are ’I’ = To
where Tr is the torque from the drive motor. Assume that J = 600.000 kgm3 B = 20,000 Nmsec. (3) Find the transfer function between the applied torque '1}; and the antenna angle 6, (b) Suppose the applied torque is computed so that 9 tracks a referenCe command 0,
according to the feedback law Tc = Kwr — 0), where K is the feedback gain. Find the transfer function between 0, and 9. Problems 163 (c) What is the maximum value of K that can be used ifyou wish to have an overshoot
M,, < 10%? (d) What values of K will provide a rise time of less than 80 sec? (Ignore the Mp
constraint.) A (9) Use MATLAB to plot the step response of the antenna system for K = 200. 400,
l000, and 2000. Find the overshoot and rise time of the four step responses by
examining your plots. Do the plots conﬁrm your calculations in parts (c) and (d)? 3.32 Show that the secondorder system
5 + 25%;, + wﬁy = 0. y(0) = yo. 540) = 0. has the response 80! 1 NU) = )‘o— sin(wd! + cos—
t/l — (2 Prove that, for the underdamped case (C < l). the response oscillations decay at a
predictable rate (see Fig. 3.63) called the logarithmic decrement C). Yo 2”; 5=ln'— :lne‘n" =ard : ——
)‘I \/l — g2
A' A = ln ii 2 ln l.
3'] )‘i
where
271 2n 1'11: — = wd (um/l — r2 is the damped natural period of vibration. The damping coefﬁcient in terms of the
logarithmic decrement is then _ 5
\/4n2 +52' C Figure 3.63
Deﬁnition of
logarithmic decrement ...
View
Full
Document
This document was uploaded on 02/22/2012.
 Fall '09

Click to edit the document details